#6890048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,35)(27,37,40,32)(28,34,29,43)(31,46,45,36)(33,44,51,42)(38,52,39,48)(41,49,56,53)(47,54,50,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,40)(26,32,35,37)(28,39,29,38)(31,41,45,56)(33,50,51,47)(34,52,43,48)(36,49,46,53)(42,55,44,54);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,29)(26,33,35,51)(27,38,40,39)(31,44,45,42)(32,47,37,50)(34,36,43,46)(41,55,56,54)(48,49,52,53);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,29,30,28)(26,34,35,43)(27,39,40,38)(31,44,45,42)(32,48,37,52)(33,36,51,46)(41,55,56,54)(47,49,50,53);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,29)(33,51)(34,43)(38,39)(42,44)(47,50)(48,52)(54,55)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6797244);